Chapter 14 Oscillations

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Chapter 14 Oscillations

1. To understand the physics and mathematics of
   oscillation.
2. To draw and interpret oscillatory graphs.
3. To learn the concepts of phase and phase constant
4. To understand and use energy conservation in
   oscillatory system
5. To understand the basic ideas of damping and
   resonance.

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Stop to think 14.1 page 414Stop to think
14.2 page 417Stop to think 14.3 page 419Stop to
think 14.4 page 423Stop to think 14.5 page 428

• Example 14.2 page 413
• Example 14.4 page 417
• Example 14.6 page 422
• Example 14.7 page 424
• Example 14.9 page 426
• Example 14.10 page 428

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Simple Harmonic Motion

• Object or systems of objects that undergo
  oscillatory motion are called oscillators. All
  these oscillators have two things in common
• 1. The oscillation takes place about an
  equilibrium position, and
• 2. The motion is periodic.

Sinusoidal oscillation Is called simple harmonic
motion.
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Period and frequency

• Period T time per cycle, units second
• Frequency f the number of cycles per second.
  units 1/s Hz (Hertz)

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Graph of simple harmonic motion

• The amplitude A the maximum displacement from
  equilibrium.
• Measured A 0.17 cm
• Measured T 1.60s
• How to describe the displacement x using A, T,
  and t.

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Position vs time graph and velocity vs time graph

• Position-vs time graph Velocity vs
  time graph

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Angular frequency ?
We define ?2p/T 2pf, is called angular
frequency
V(max) ?A
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Simple Harmonic Motion and Circular Motion.

• Uniform circular motion projected onto one
  dimension is simple harmonic motion
• The figure shows the x-component, when the
  particle does uniform circular motion
• With
• So

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The Phase constant

• In more general case, particle start phase Fo is
  not zero. then,
• The harmonic motion function is
• Fo is called the phase constant or initial phase.
• is called
  phase.
• When t 0, initial condition

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Show phase constant

• The following show the oscillations by different
  phase constant

Notice Fop/3 and Fo-p/3 have the same starting
x, but different Vo
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P14.2

• From the Figure, how we get
• Amplitude
• Frequency
• Phase constant.
• First, you write general Harmonic
• Wave function
• Then you compare this trigonometric
• Function and the figure, you can get
• A 10 cm
• T 2 s, frequency f ½ 0.5 Hz.
• When t 0 x(0) 5cm 10cos(Fo)
• cos(Fo)0.5, Fop/3.
• But at t 0, the slope of curve is negative
• So V0 is negative, from
• Sin(Fo) is positive, that makes Fop/3.

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Energy in simple Harmonic Motion

• The mechanical energy of an object oscillating on
  a spring is
• When x A, E ½ kA2 0
• When x 0 E 0 1/2mV2 max
• From conservation of energy

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The Dynamics of Simple Harmonic Motion

• The spring force is
• From Newtons second Law
• The dynamics equation
• This is second derivative equation, the solution
  is

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Vertical oscillations

• The equilibrium position, ?L.
• The harmonic oscillation equation should be the
  same on a horizontal spring.
• In right figure
• K 10N/m, The spring stretch at equilibrium
  is given by ?Lmg / K 19.6 cm
• That is the amplitude of oscillation
• A 30cm-19.6cm 10.4 cm
• The initial condition y0-A AcosFo
• Fop. So the oscillator function is

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The Pendulum

• Lets look another oscillator a pendulum

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Small-angle Approximation

• The Dynamical equation is
• Using
• We can write
• If ? is very small sin (?) ? (? in radians)
• Then
• Solution is
• Or

or
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The Physical pendulum
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Period vs Meff in log plot
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V-10p sin(ptp/3)